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THÔNG TIN TÀI LIỆU
Cấu trúc
Acknowledgments
Summary
List of Tables
List of Figures
List of Abbreviations
List of Symbols
1 Introduction
1.1 Quantum state estimation
1.2 Symmetric informationally complete POMs
2 Quantum state estimation
2.1 Introduction
2.2 Historical background
2.3 Quantum states and measurements
2.3.1 Simple systems
2.3.2 Composite systems
2.4 Quantum state reconstruction
2.4.1 Linear state reconstruction
2.4.2 Maximum-likelihood estimation
2.4.3 Other reconstruction methods
2.5 Fisher information and Cramér--Rao bound
3 Fully symmetric measurements and product measurements
3.1 Introduction
3.2 Setting the stage
3.2.1 Linear state tomography
3.2.2 Tight IC measurements
3.3 Applications of random-matrix theory to quantum state tomography
3.3.1 A simple idea
3.3.2 Isotropic measurements
3.3.3 Tight IC POMs and SIC POMs
3.3.4 Qubit tomography
3.4 Joint SIC POMs and Product SIC POMs
3.4.1 Bipartite scenarios
3.4.2 Multipartite scenarios
3.5 Summary
4 The power of informationally overcomplete measurements
4.1 Introduction
4.2 Optimal state reconstruction
4.2.1 Optimal reconstruction in the perspective of frame theory
4.2.2 Connection with the maximum-likelihood method
4.3 Quantum state estimation with mutually unbiased measurements
4.4 Efficiency of covariant measurements
4.5 Informationally overcomplete measurements on the two-level system
4.6 Summary
5 Optimal state estimation with adaptive measurements
5.1 Introduction
5.2 Quantum Fisher information and quantum CR bound
5.2.1 One-parameter setting
5.2.2 Multiparameter setting
5.3 Gill--Massar trace and Gill--Massar bound
5.3.1 Reexamination of the Gill--Massar inequality
5.3.2 Gill--Massar bound for the scaled WMSE
5.3.3 Gill--Massar bounds for the mean square Bures distance and the mean square HS distance
5.4 Optimal quantum state estimation with adaptive measurements
5.4.1 A general recipe
5.4.2 Approximate saturation of the Gill--Massar bound for the MSB
5.4.3 Degenerate two-level systems
5.4.4 Comparison with nonadaptive schemes
5.5 Summary and open problems
6 Quantum state estimation with collective measurements
6.1 Introduction
6.2 Efficiency of asymptotic state estimation
6.2.1 Quantum Cramér--Rao bound based on the right logarithmic derivative
6.2.2 Efficiency of the optimal state estimation in the asymptotic limit
6.3 Quantum state estimation with coherent measurements
6.3.1 Schur--Weyl duality and its implications
6.3.2 Highest-weight states and coherent states
6.3.3 Coherent measurements
6.3.4 Complementarity polynomials
6.3.5 Estimation of highly mixed states with collective measurements
6.4 Collective measurements in qubit state estimation
6.4.1 A lower bound for the weighted mean square error
6.4.2 Fisher information matrices for coherent measurements
6.4.3 Complementarity polynomials
6.4.4 Mean square error and mean square Bures distance
6.5 Summary and open problems
7 Symmetric informationally complete POMs
7.1 Introduction
7.2 Symmetry and group covariance
7.2.1 Groups that can generate SIC POMs
7.2.2 Orbits and equivalence of SIC POMs
7.3 Heisenberg--Weyl group and Clifford group
7.3.1 Heisenberg--Weyl group
7.3.2 Special linear group
7.3.3 Understanding the Clifford group from a homomorphism
8 SIC POMs in prime dimensions
8.1 Introduction
8.2 Group covariant SIC POMs are HW covariant
8.3 Qubit SIC POMs
8.4 SIC POMs in prime dimensions not equal to 3
8.5 SIC POMs in dimension 3
8.5.1 Symmetry of SIC POMs
8.5.2 Infinitely many inequivalent SIC POMs
8.6 Beyond prime dimensions
8.7 Summary
9 Two-qubit SIC POMs
9.1 Introduction
9.2 Structure of SIC POMs in the four-dimensional Hilbert space
9.2.1 Symmetry transformations within an HW covariant SIC POM
9.2.2 Symmetry transformations among HW covariant SIC POMs
9.2.3 SIC POM regrouping phenomena
9.3 Two-qubit SIC POMs
9.3.1 Two-qubit SIC POMs in the product basis
9.3.2 Two-qubit SIC POMs in the Bell basis
9.4 Summary
10 Symmetry and equivalence
10.1 Introduction
10.2 SIC POMs and graph automorphism problem
10.2.1 Unitary symmetry and permutation symmetry
10.2.2 A connection with the graph automorphism problem
10.2.3 An algorithm
10.3 HW covariant SIC POMs
10.3.1 SIC POMs in dimension 3 revisited
10.3.2 Symmetry and equivalence
10.3.3 Nice error bases in the symmetry group
10.4 Hoggar lines
10.5 Quest for new SIC POMs
10.6 Summary and open questions
A Several distance and distinguishability measures
A.1 Hilbert--Schmidt distance and trace distance
A.2 Fidelity and Bures distance
B Weighted t-designs
C Proof of Lemma 4.1
D Supplementary materials about adaptive measurements
D.1 Derivation of Eqs. (5.23) and (5.24)
D.2 Connection with pure-state estimation
D.3 Discontinuity of the minimal scaled MSB
E Technical details about Chapter 6
E.1 Proof of Eq. (6.49) for Slater-determinant states
E.2 Proof of Conjecture 6.4 for symmetric and antisymmetric subspaces
E.3 Proof of Theorem 6.7
E.4 Proof of Theorem 6.8
F Nice error bases
G p-groups and Sylow's theorem
H Supplementary information about the Clifford group
H.1 Trace of a Clifford unitary operator
H.2 Normalizer of the Clifford group
H.3 HW groups in the Clifford group in a prime dimension
I Some basic concepts in graph theory
Bibliography
Index
Nội dung
Quantum State Estimation and Symmetric Informationally Complete POMs ZHU HUANGJUN NATIONAL UNIVERSITY OF SINGAPORE 2012 Quantum State Estimation and Symmetric Informationally Complete POMs ZHU HUANGJUN (M.Sc., Peking University) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY NUS GRADUATE SCHOOL FOR INTEGRATIVE SCIENCES AND ENGINEERING CENTRE FOR QUANTUM TECHNOLOGIES NATIONAL UNIVERSITY OF SINGAPORE 2012 Acknowledgments I am sincerely grateful to my supervisor Berthold-Georg Englert for his guidance, for giving me the opportunities and freedom to explore various interesting topics, and for encouraging me to attend conferences and make presentations. I would also like to thank Markus Grassl for numerous stimulating discussions, especially those on symmetric informationally complete probability operator measurements (SIC POMs), and for critical comments and suggestions on writing. Special thanks to Chen Lin, Masahito Hayashi, Teo Yong Siah, Wei Tzu-Chieh, Jaroslav Řeháček, Zdeněk Hradil, Valerio Scarani, Cyril Branciard, and Xu Aimin for fruitful discussions and collaborations. I am also grateful to Marcus Appleby, Christopher Fuchs, Ingemar Bengtsson, and Andreas Winter for discussions and encouragement. Special thanks also to Yao Penghui, Ng Hui Khoon, Amir Kalev, Philippe Raynal, Tan Si-Hui, Lü Xin, Wang Guangquan, Thiang Guo Chuan, Tomasz Paterek, Tomasz Karpiuk, Ma Jia Jun, Kwek Leong Chuan, Thomas Durt, Daniel Greenberger, and Andrew Scott for stimulating discussions. I would also like to thank Wang Jian-Sheng and Gong Jiangbin for serving on my thesis advisory committee. Special thanks to Dai Li and Lee Kean Loon for enthusiastic help related to the format of the thesis. Special thanks also to the examiners for reviewing this thesis and for providing generous comments and suggestions. I would like to acknowledge the financial support from NUS Graduate School for Integrative Sciences and Engineering (NGS), and I am grateful to many administrative staff for their dedication to the welfare of students. I am also grateful to Centre for Quantum Technologies and many administrative staff for providing a comfortable research environment and numerous timely help. Finally, many thanks to my parents and my sister for their continuous support and understanding, and to my friends for their encouragement. During the PhD candidature, I have mainly worked on three related topics: quantum state estimation, SIC POMs, and multipartite entanglement. This thesis covers the main results concerning the first two topics, most of which have not been published. Chapters 3, 8, and are based on the following three papers, respectively: ii • H. Zhu and B.-G. Englert, Quantum state tomography with fully symmetric measurements and product measurements, Phys. Rev. A 84, 022327 (2011). • H. Zhu, SIC POVMs and Clifford groups in prime dimensions, J. Phys. A: Math. Theor. 43, 305305 (2010). • H. Zhu, Y. S. Teo, and B.-G. Englert, Two-qubit symmetric informationally complete positive-operator-valued measures, Phys. Rev. A 82, 042308 (2010). Other papers not covered in this thesis: • Y. S. Teo, H. Zhu, B.-G. Englert, J. Řeháček, and Z. Hradil, Quantum-state reconstruction by maximizing likelihood and entropy, Phys. Rev. Lett. 107, 020404 (2011). • L. Chen, H. Zhu, and T.-C. Wei, Connections of geometric measure of entanglement of pure symmetric states to quantum state estimation, Phys. Rev. A 83, 012305 (2011). • H. Zhu, L. Chen, and M. Hayashi, Additivity and non-additivity of multipartite entanglement measures, New J. Phys. 12, 083002 (2010). • L. Chen, A. Xu, and H. Zhu, Computation of the geometric measure of entanglement for pure multiqubit states, Phys. Rev. A 82, 032301 (2010). • C. Branciard, H. Zhu, L. Chen, and V. Scarani, Evaluation of two different entanglement measures on a bound entangled state, Phys. Rev. A 82, 012327 (2010). • H. Zhu, Y. S. Teo, and B.-G. Englert, Minimal tomography with entanglement witnesses, Phys. Rev. A 81, 052339 (2010). • Y. S. Teo, H. Zhu, and B.-G. Englert, Product measurements and fully symmetric measurements in qubit-pair tomography: A numerical study, Opt. Commun. 283, 724 (2010). iii Contents Acknowledgments ii Summary x List of Tables xi List of Figures xii List of Abbreviations xiv List of Symbols xv Introduction 1.1 Quantum state estimation . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Symmetric informationally complete POMs . . . . . . . . . . . . . . . . Quantum state estimation 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 Historical background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.3 Quantum states and measurements . . . . . . . . . . . . . . . . . . . . . 18 2.4 iv 11 2.3.1 Simple systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.3.2 Composite systems . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Quantum state reconstruction . . . . . . . . . . . . . . . . . . . . . . . . 21 2.4.1 Linear state reconstruction . . . . . . . . . . . . . . . . . . . . . 22 2.4.2 Maximum-likelihood estimation . . . . . . . . . . . . . . . . . . . 23 2.4.3 Other reconstruction methods . . . . . . . . . . . . . . . . . . . . 25 Contents 2.5 Fisher information and Cramér–Rao bound . . . . . . . . . . . . . . . . 27 Fully symmetric measurements and product measurements 31 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.2 Setting the stage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.3 3.4 3.5 3.2.1 Linear state tomography . . . . . . . . . . . . . . . . . . . . . . . 34 3.2.2 Tight IC measurements . . . . . . . . . . . . . . . . . . . . . . . 36 Applications of random-matrix theory to quantum state tomography . . 38 3.3.1 A simple idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.3.2 Isotropic measurements . . . . . . . . . . . . . . . . . . . . . . . 40 3.3.3 Tight IC POMs and SIC POMs . . . . . . . . . . . . . . . . . . . 42 3.3.4 Qubit tomography . . . . . . . . . . . . . . . . . . . . . . . . . . 45 Joint SIC POMs and Product SIC POMs . . . . . . . . . . . . . . . . . 50 3.4.1 Bipartite scenarios . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.4.2 Multipartite scenarios . . . . . . . . . . . . . . . . . . . . . . . . 53 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 The power of informationally overcomplete measurements 57 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.2 Optimal state reconstruction . . . . . . . . . . . . . . . . . . . . . . . . 59 4.2.1 Optimal reconstruction in the perspective of frame theory . . . . 60 4.2.2 Connection with the maximum-likelihood method . . . . . . . . . 63 4.3 Quantum state estimation with mutually unbiased measurements . . . . 64 4.4 Efficiency of covariant measurements . . . . . . . . . . . . . . . . . . . . 68 4.5 Informationally overcomplete measurements on the two-level system . . 72 4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 Optimal state estimation with adaptive measurements 77 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 5.2 Quantum Fisher information and quantum CR bound . . . . . . . . . . 80 5.2.1 One-parameter setting . . . . . . . . . . . . . . . . . . . . . . . . 80 v Contents 5.2.2 5.3 Multiparameter setting . . . . . . . . . . . . . . . . . . . . . . . 84 Gill–Massar trace and Gill–Massar bound . . . . . . . . . . . . . . . . . 85 5.3.1 Reexamination of the Gill–Massar inequality . . . . . . . . . . . 86 5.3.2 Gill–Massar bound for the scaled WMSE . . . . . . . . . . . . . 87 5.3.3 Gill–Massar bounds for the mean square Bures distance and the mean square HS distance . . . . . . . . . . . . . . . . . . . . . . 89 5.4 5.5 Optimal quantum state estimation with adaptive measurements . . . . . 92 5.4.1 A general recipe . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 5.4.2 Approximate saturation of the Gill–Massar bound for the MSB . 100 5.4.3 Degenerate two-level systems . . . . . . . . . . . . . . . . . . . . 103 5.4.4 Comparison with nonadaptive schemes . . . . . . . . . . . . . . . 107 Summary and open problems . . . . . . . . . . . . . . . . . . . . . . . . 110 Quantum state estimation with collective measurements 113 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 6.2 Efficiency of asymptotic state estimation . . . . . . . . . . . . . . . . . . 115 6.2.1 Quantum Cramér–Rao bound based on the right logarithmic derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 6.2.2 6.3 6.4 vi Efficiency of the optimal state estimation in the asymptotic limit 118 Quantum state estimation with coherent measurements . . . . . . . . . . 122 6.3.1 Schur–Weyl duality and its implications . . . . . . . . . . . . . . 123 6.3.2 Highest-weight states and coherent states . . . . . . . . . . . . . 125 6.3.3 Coherent measurements . . . . . . . . . . . . . . . . . . . . . . . 127 6.3.4 Complementarity polynomials . . . . . . . . . . . . . . . . . . . . 128 6.3.5 Estimation of highly mixed states with collective measurements . 133 Collective measurements in qubit state estimation . . . . . . . . . . . . . 136 6.4.1 A lower bound for the weighted mean square error . . . . . . . . 137 6.4.2 Fisher information matrices for 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Gen., 34:7111, 2001. 274 Index D-invariant, 117 p-group, 246 t-design, 232 weighted, 232 2-design mutually unbiased bases (MUB), 232 SIC POM, 232 weighted, 38 SLD Fisher information, 82, 231 canonical form of a graph, 255 canonical order-3 Clifford unitary transformation, 153 central-limit theorem, quantum, 16, 114 Clifford group, 153, 160, 247 normalizer of, 190, 251 structure, 163 adaptive measurement, 5, 77 coherent measurement, 6, 127 comparison with nonadaptive complementarity polynomial, 128, 144 schemes, 107 covariant, 142, 148 degenerate two-level system, 103 GMT, 131, 144 optimal quantum state estimation, 92 optimal, 148 two-step adaptive, 82 quantum state estimation, 122 adjacency matrix, 255 qubit state estimation, 140, 146 angle matrix, 203, 212 coherent state, 6, 126 angle tensor, 203 conjecture, 132 Appleby, 153, 183 collective measurement, 6, 20 Clifford group, 153, 164 asymptotic state estimation, 115 equivalence criterion for SIC POMs, coherent measurement, see coherent 155, 204 measurement HW group, 161 quantum state estimation, 113 SIC POMs in dimension three, 176 qubit state estimation, 136 automorphism, 256 collineation group, 156 automorphism group, 256 commutation superoperator, 117 complementarity polynomial, 128, 131 balanced measurement, 67 qubit, 144 SIC POM and MUB, 67 complementarity principle, 1, 85 Bayesian approach, 17 conjecture Bayesian mean estimation (BME), 25 balanced measurement, 67 Born rule, 18 coherent state, 132 bound extremal Fisher information matrix, Cramér–Rao (CR), 27 97 generalized Gill–Massar (GM), 137 Gill–Massar trace (GMT), 132 Gill–Massar (GM), 85 HW covariant SIC POM, 181 joint, 138 quantum Cramér–Rao (CR), 80, 115 conjecture of Slater, 145 RLD, 115 Zauner, 153, 154 Scott, 38 covariant measurement, 68 SLD, 81 Welch, 152 covariant coherent measurement, 142, Bures distance, 231 148 275 Index Cramér–Rao (CR) bound, 2, 27, see product measurement, 51 also quantum Cramér–Rao (CR) Gaussian bound approximation, 36 displacement operator, 165 unitary ensemble, 39 general linear group, 123 entangled measurement, 21 generalized Bloch vector (GBV), 192 entangled state, 20 generalized Gill–Massar (GM) bound, 137 entanglement, 20 generalized measurement, 18 equiangular lines, 152 generalized Pauli group, see HW group Welch bound, 152 geometric phase, 178, 203 equivalence problem, 8, 156, 200 Gill–Massar (GM), 15 extended automorphism group, 256 bound, extended Clifford group, 163 generalized, 137 extended special linear group, 163 qubit, 88 extended symmetry group, 157 scaled MSB, 89 scaled MSH, 90, 91 Fano, 11, 22 scaled WMSE, 88 fidelity, 1, 17, 230 inequality, 87 fiducial measurement, 31 theorem, 86 fiducial POM, trace, see GMT fiducial state, GMT, 5, 86 dimension eight, 224 asymptotic maximum, 121, 135, 144 dimension four, 184 coherent measurement, 131 dimension three, 176 complementarity polynomial, 131 Hoggar lines, 221 maximum in qubit state estimation, known solutions, 152 145 qubit, 168 maximum over collective measurefigures of merit ments when ρ = 1/d, 135 Bures distance, 231 Slater’s conjecture, 145 fidelity, 230 graph, 205, 255 Hilbert–Schmidt (HS) distance, 229 graph automorphism problem, 202, 256 trace distance, 229 graph isomorphism problem, 202, 256 Fisher information, 27 graph-theoretic approach, 10, 200 Fisher information matrix, 2, 28 Grassl, 153–155, 223 Uρ -invariant, 93, 103 group coherent measurement, 128, 140 p-group and Sylow p-subgroup, 246 extremal, 95 (extended) Clifford group, see Clifford frame superoperator, 63 group GM inequality, 86 (extended) automorphism group, 256 mutually unbiased measurements on a (extended) special linear group, 163 qubit, 75 (extended) symmetry group, 157 RLD bound, 116, 120 general linear group, 123 SLD bound, 81 generalized Pauli group, see HW frame superoperator, 34, 35 group covariant measurement, 68 Fisher information matrix, 63 Heisenberg–Weyl (HW) group, see optimal reconstruction, 60 HW group 276 Index index group, 245 tight, 31, 36 informationally overcomplete measuresymmetric group, 123 ment, 4, 20, 57, 59 three-qubit Pauli group, 162 isomorphism, 255 group covariance, 151, 157 groups that can generate SIC POMs, isotropic measurement, 40, 41, 45 Ivanović, 12, 58 159, 168 Hayashi, 15, 16, 114 hedged maximum-likelihood estimation (HMLE), 13, 26 height of a partition, 123 Heisenberg–Weyl group, see HW group Helstrom, 11, 15, 81 highest-weight state, 125 Hilbert–Schmidt (HS) distance, 1, 229 Hoggar lines, 9, 10, 155, 162, 196, 226 fiducial state, 221 group covariance and symmetry, 180, 220 Holevo, 15 bound, 15, 16, 117 commutation superoperator, 117 lemma, 116 homogeneous complementarity polynomial, 131 Hradil, 13, 24, 58 HW covariant SIC POM, 211 beyond prime dimensions, 180 dimension three, 171, 212 equivalence relation, 171, 175, 177, 181, 215 prime dimensions, 8, 167, 169 qubit, 168 regrouping phenomenon, 189 symmetry group, 171, 175, 178, 180, 185, 215 two-qubit, 9, 183, 191 HW group, 7, 160 in the Clifford group, 252, 253 standard, 176, 253 hypergraph, 205, 256 k-uniform, 256 index group, 245 individual measurement, 20 informationally complete (IC), 3, 19 minimal, 20 joint bound, 138 joint SIC POM, 4, 32, 50 lemma N -partite symmetric state, 241 a matrix inequality, 60, 233 bipartite antisymmetric state, 242 Gill–Massar trace (GMT), 132 HW covariant SIC POM, 169 orbits of states in a SIC POM, 158 SLD Fisher information matrix, 129 lemma of Holevo, 116 likelihood functional, 23 linear state reconstruction, 22 linear state tomography, 3, 34 local asymptotic normality, 16, 114 log likelihood functional, 23 maximum entropy (ME) principle, 13 maximum-likelihood (ML) estimation, 13, 23 estimator, 24, 28 method, 63 principle, 23 mean square Bures distance (MSB), 2, see also scaled MSB mean square error (MSE), 31, see also scaled MSE matrix, 28 weighted, mean square Hilbert Schmidt distance (MSH), 1, see also scaled MSH measurement, 18 adaptive measurement, 77 coherent measurement, see coherent measurement collective measurement, see collective measurement covariant measurement, 68 entangled measurement, 21 individual measurement, 20 277 Index informationally complete (IC) measurement, 19 informationally overcomplete measurement, 20, 57 isotropic measurement, 40 mutually unbiased measurements, see mutually unbiased measurements probability operator measurement (POM), 19 product measurement, 21, 50 separable measurement, 21 symmetric informationally complete (SIC) measurement, 3, see also SIC POM tight informationally complete (IC) measurement, 36 measurement operator, 18 monomial, 170 monotone Riemannian metric, 71, 78 mutually unbiased bases (MUB), 3, 64 mutually unbiased measurements, 3, 12, 58, 64 balanced measurement, 67 MSE matrix, 66 optimality in state estimation, 66 qubit state estimation, 48, 74, 89 Nagaoka, 15, 16 nice error basis, 245 SIC POM, 159, 180, 216, 223, 226 nice graph, 210 normalizer of the Clifford group, 190, 251 open questions, see also conjecture quantum state estimation, 111, 150 SIC POM, 227 ordered partition, 206 product SIC POM, 4, 32 bipartite, 50 multipartite, 54 projective measurement, 19 pure-state limit adaptive measurement, 235, 238 asymptotic state estimation, 120, 121 coherent measurement, 143, 145, 147 covariant measurement, 71 pure-state models, 16 quantum central-limit theorem, 16, 114 quantum cloning, 17 quantum Cramér–Rao (CR) bound, 15, 80, see also SLD bound and RLD bound quantum estimation theory, 11, 78, 114 quantum Fisher information, 80, see also SLD Fisher information and RLD Fisher information quantum process tomography (QPT), 14 quantum state estimation, adaptive measurement, 77 collective measurement, 113 asymptotic limit, 115 coherent measurement, 122 nonadaptive measurement linear reconstruction, 31 optimal reconstruction, 57 open problems, 111, 150 quantum state tomography, see quantum state estimation qubit state estimation adaptive measurement, 88 collective measurement, 136 nonadaptive measurement linear reconstruction, 45 optimal reconstruction, 72 quorum, 11 partition, 123 height, 123 ordered, 206 random-matrix theory, 3, 38 Pauli-exclusion principle, 134 positive-operator-valued measure reconstruction operator, 35, 201 canonical, 35 (POVM), see probability opinformationally incomplete measureerator measurement (POM) ment, 62 probability operator measurement (POM), linear state reconstruction, 22 1, 19 product measurement, 21, 50 optimal, 60 278 Index product measurement, 51 tight IC measurement, 37 recursive ordered partition (ROP), 208 reference vertex, 206 regrouping phenomenon of SIC POMs, 9, 189 Renes, 153, 155 right logarithmic derivative (RLD), 15, 116 RLD bound, 116 scaled GMT, 121 scaled MSB, 120 scaled MSE, 120 scaled WMSE, 116 RLD Fisher information, 116, 118, 119 ROP, 208 scaled mean HS distance, 40, 42, 43 scaled mean trace distance, 39 isotropic measurement, 42, 48 product SIC POM, 52, 54 qubit tomography, 48 SIC POM, 43 scaled MSB approximate saturation of the GM bound, 100 covariant measurement, 70 discontinuity at the boundary of the state space, 237 GM bound, 89 optimal adaptive measurement, 109 qubit state estimation with collective measurements, 146 RLD bound (saturated in the asymptotic limit), 120 scaled MSE covariant measurement, 70 linear reconstruction, 35 mutually unbiased measurements, 66 optimal reconstruction, 61 product SIC POM, 51, 54 qubit, 46, 73, 74 RLD bound (saturated in the asymptotic limit), 120 tight IC measurement, 37 scaled MSE matrix, 28, 35, 120 qubit, 46, 75 scaled MSH GM bound, 90, 91 optimal adaptive measurement, 108 qubit state estimation with collective measurements, 146 Schur symmetric polynomial, 124, 140 Schur–Weyl duality, 123 Scott bound, 38 SIC POM, 153, 154 tight IC measurement, 3, 31, 36 separable measurement, 21 GMT, 86 separable state, 20 Sibley’s theorem, 170 SIC POM, 7, 151 equiangular lines, 152 equivalence or equivalent, 153, 159 equivalence problem, 8, 156, 200 fiducial state, see fiducial state graph-theoretic approach, 200 group covariance, 151, 157, 159, 168 Hoggar lines, see Hoggar lines HW covariant, see HW covariant SIC POM known solutions, 152 MUB, 49, 67, 74, 151 nice error basis, 159, 180, 216, 223, 226 numerical search, 153, 154, 225 open questions, 67, 227 orbit, 153, 159 quantum state estimation, 42, 48, 50, 74 symmetry group, 157 symmetry problem, 8, 156, 200 algorithm, 206 tight IC measurement, 31, 38 unitary symmetry and permutation symmetry, 201 skew isomorphism, 255 Slater’s conjecture, 145 Slater-determinant state, 126, 240, 242 SLD bound, 81 equivalent formulations, 83 scaled MSB, 85 scaled MSH, 85 scaled WMSE, 84 279 Index SLD Fisher information, 81, 84, 118, 119 Bures distance, 82 special linear group, 163 standard HW group, 176, 253 state reconstruction, 21 Bayesian mean estimation (BME), 25 hedged maximum-likelihood estimation (HMLE), 26 linear state reconstruction, 22, 34 maximum-likelihood estimation (MLE), 23 optimal reconstruction in the perspective of frame theory, 60 strong group covariant, 157 swap operator, 124, 129 Sylow p-subgroup, 246 group covariant SIC POM, 169, 171, 180 Sylow’s theorem, 246 symmetric group, 123 symmetric informationally complete (SIC), 3, 151 symmetric informationally complete probability operator measurement, see SIC POM symmetric logarithmic derivative (SLD), 11, 81, 83 symmetry group of a SIC POM, 157 symmetry problem, 8, 156, 200 symplectic operator, 165 prime dimensions not equal to three, 171 trace of a Clifford operator, 248 theorem of Gill–Massar, 86 Sibley, 170 Sylow, 246 Zauner, 158 three-qubit Pauli group, 162 tight informationally complete (IC), 31, 36 quantum state estimation, 42 weighted 2-design, 38 trace distance, 1, 32, 229 triple product, 178, 185, 203 two-step adaptive, 82 unimodular, 157, 170 von Neumann measurement, see projective measurement weighted t-design, 232 weighted 2-design, 31, 38 weighted 3-design, 41 weighted mean square error (WMSE), 2, 29 convex optimization, 99 generalized GM bound, 137, 138 GM bound, 87 joint bound, 138, 139 RLD bound, 116 Welch bound, 152 Wigner function, 12, 13, 151 theorem Wigner semicircle law, 39 asymptotic Fisher information matrix, Wootters, 12, 17, 58, 64, 113 143, 244 Zauner, 152, 220 asymptotic GMT, 135, 243 conjecture, 153, 154 Gill–Massar trace (GMT), 133 matrix, 165 group covariant SIC POM in a prime theorem, 157 dimension, 168 unitary transformation, 153, 165 groups that can generate SIC POMs, zero-eigenvalue problem, 25, 26 159 normalizer of the Clifford group, 252 orbits and equivalence of SIC POMs, 159 symmetry of a SIC POM beyond prime dimensions, 180 dimension three, 175 280 [...]... 254 xviii Chapter 1 Introduction 1.1 Quantum state estimation Quantum state estimation is a procedure for inferring the state of a quantum system from generalized measurements, known as probability operator measurements (POMs) It is a primitive of many quantum information processing tasks, such as quantum computation, quantum communication, and quantum cryptography, because all these tasks... quantum information science: quantum state estimation and symmetric informationally complete probability operator measurements (SIC POMs) 1 Part I of this thesis focuses on reliable and efficient estimation of mixed states of finite-dimensional quantum systems in the large-sample scenario Four natural settings are investigated in the order of sophistication levels: independent and identical measurements... relation, and the geometry of quantum states, from the information-theoretic perspective Chapter 2 presents an overview of quantum state estimation from the theoretical perspective We start with a historical survey of the major achievements in the field during the past half a century and then introduce several basic ingredients in quantum state estimation, such as quantum states, measurements, state reconstruction,... inspiration from quantum state tomography Standard QPT (SQPT) was introduced by Poyatos, Cirac, and Zoller [221], as well as by Chuang and Nielsen [66] in the late 1990s To characterize a quantum process, a set of reference states is prepared and then reconstructed by quantum state tomography after subjecting them to a given quantum process, which can then be determined if the set of reference states spans... elements in the field of quantum state estimation, such as quantum states, measurements, state reconstruction, Fisher information, and CR bound 2.2 Historical background The idea of determining the state of a quantum system from measurements can be traced back to Pauli when he asked whether the position distribution and momentum distribution suffice to determine the wave function of a quantum system [211]... respect to the scaled MSH of the standard state estimation, state estimation with covariant measurements, and state estimation with optimal adaptive measurements 108 5.3 Tomographic efficiencies with respect to the scaled MSB of covariant measurements and optimal adaptive measurements 109 6.1 Contour plots of the asymptotic maximal scaled MSE, MSB, and the minimal scaled GMT in the... 6 1.2 Symmetric informationally complete POMs N identically prepared quantum systems It has profound implications for understanding the tomographic efficiencies and distinctive features of collective measurements It is useful not only for determining the efficiency gap between separable measurements and collective measurements but also for explicating the emergence of universality in optimal state estimation. .. approaches, especially in the study of quantum metrology [108, 109, 110] As the requirement for precision measurements increases, the integration of the two approaches is due to play an increasingly important role 11 Chapter 2 Quantum state estimation In this chapter, we first present a historical survey of the development of quantum estimation theory and quantum state estimation in particular We then introduce...Contents 6.5 Summary and open problems 149 7 Symmetric informationally complete POMs 151 7.1 Introduction 151 7.2 Symmetry and group covariance 156 7.2.1 7.2.2 7.3 Groups that can generate SIC POMs 157 Orbits and equivalence of SIC POMs 159 Heisenberg–Weyl group and Clifford group ... Following their experiment, states of many other quantum systems were also characterized, such as the vibrational state of a diatomic molecule [82], the motional state of a trapped ion [174], the state of an ensemble of helium atoms [171], and entangled states of polarized photon pairs [156, 267] See Refs [186, 208] for an overview about experimental progress in quantum state estimation The advance of . Quantum State Estimation and Symmetric Informationally Complete POMs ZHU HUANGJUN NATIONAL UNIVERSITY OF SINGAPORE 2012 Quantum State Estimation and Symmetric Informationally Complete POMs ZHU. Introduction 1 1.1 Quantum state estimation . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Symmetric informationally complete POMs . . . . . . . . . . . . . . . . 7 2 Quantum state estimation. in quantum information science: quantum state estimation and symmetric informationally complete probability operator measurements (SIC POMs) 1 . Part I of this thesis focuses on reliable and